significant figures

Significant FiguresSignificant Figures

In MeasurementsIn Measurements

Significant FiguresSignificant FiguresAt the conclusion of our time At the conclusion of our time

together, you should be able to:together, you should be able to:

1. Explain what significant figures are in a measurement

2. Determine the number of significant figures in any measurement


The significant figures in a measurement include all The significant figures in a measurement include all of the digits that are known, plus one last digit of the digits that are known, plus one last digit that is that is estimatedestimated..

The numbers reported in a measurement are The numbers reported in a measurement are limited by the measuring limited by the measuring tool.tool.

How many sig figs are there in a given How many sig figs are there in a given measurement?measurement?

Measurement and Significant FiguresMeasurement and Significant Figures

Every experimental Every experimental measurement has a degree measurement has a degree of of uncertaintyuncertainty..

The volume, V, at right is The volume, V, at right is certain in the 10’s place, certain in the 10’s place, 10mL<V<20mL10mL<V<20mL

The 1’s digit is also certain, The 1’s digit is also certain, 17mL<V<18mL17mL<V<18mL

A best guess is needed for A best guess is needed for the the tenthstenths place. place.

To indicate the precision of a measurement, To indicate the precision of a measurement, the value recorded should use all the digits the value recorded should use all the digits known with certainty, plus one additional known with certainty, plus one additional estimated digit that usually is considered estimated digit that usually is considered uncertain by plus or minus uncertain by plus or minus 11..

NoNo further insignificant digits should be further insignificant digits should be recorded. recorded.

The total number of digits used to express The total number of digits used to express such a measurement is called the number of such a measurement is called the number of significant figures.significant figures.

All but one of the significant figures are All but one of the significant figures are known with certainty. The last significant known with certainty. The last significant figure is only the best possible figure is only the best possible estimateestimate..

Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of

precision or certainty.

1. assume a = b2. multiply by b

3. ab = b2

4. subtract a2

5. ab – a2 = b2 – a2

6. factor each7. a(b – a) = (b + a)(b – a)8. divide by (b – a)9. a = 2a10. divide by a11. therefore: 1 = 2

Does 1 = 2???

An Easier Way to have 1 = 2

Reading a MeterstickReading a Meterstick

. l. l22. . . . I . . . . I. . . . I . . . . I33 . . . .I . . . . I . . . .I . . . . I44. . cm. . cm

First digit (known)First digit (known) = 2 = 2 2.?? cm2.?? cm

Second digit (known)Second digit (known) = 0.7 = 0.7 2.7? cm2.7? cm

Third digit (estimated) between Third digit (estimated) between 0.03- 0.050.03- 0.05

Length reportedLength reported == 2.74 cm 2.74 cm

oror 2.73 cm 2.73 cm

oror 2.75 cm2.75 cm

Known + Estimated DigitsKnown + Estimated Digits

In 2.74 cm…In 2.74 cm…

• Known digitsKnown digits 2 andand 7 are 100% certainare 100% certain

• The third digit The third digit 44 is estimated (uncertain) is estimated (uncertain)

• In the reported length, all In the reported length, all threethree digits digits (2.74 cm) are significant including the (2.74 cm) are significant including the estimated oneestimated one

Learning CheckLearning Check

. l. l88. . . . I . . . . I. . . . I . . . . I99. . . . I . . . . I. . . . I . . . . I1010. . cm. . cm

What is the length of the line?What is the length of the line?

1) 9.3 cm 1) 9.3 cm

2) 9.32 cm 2) 9.32 cm

3) 9.33 cm3) 9.33 cm

How does your answer compare with your How does your answer compare with your

neighbor’s answer? Why or why not?neighbor’s answer? Why or why not?

Zero as a Measured NumberZero as a Measured Number

. l. l33. . . . I . . . . I. . . . I . . . . I44 . . . . I . . . . I . . . . I . . . . I55. . cm. . cm

What is the length of the line?What is the length of the line?

First digitFirst digit 55.?? cm.?? cm

Second digitSecond digit 5.05.0? cm? cm

Last (estimated) digit isLast (estimated) digit is 5.005.00 cmcm

Always estimate ONE place past the Always estimate ONE place past the smallest mark!smallest mark!

11.5 mL

So how many sig figs are there in a given So how many sig figs are there in a given measurement?measurement?

52.8 mL

Beware of Identity Theft!!

How to Determine Significant Figures in a How to Determine Significant Figures in a ProblemProblem

Use the following rules:Use the following rules:

Rule #1Rule #1

Every Every nonzerononzero digit is significant digit is significant

Examples:Examples:

24 = 224 = 2

3.56 = 33.56 = 3

7 = 17 = 1

Rule #2 – Sandwiched 0’sRule #2 – Sandwiched 0’s

ZerosZeros between non-zeros are significant between non-zeros are significant

Examples:Examples:

7003 = 47003 = 4

40.9 = 40.9 = 3

Rule #3 – Leading 0’sRule #3 – Leading 0’s

Zeros appearing in Zeros appearing in frontfront of non-zero digits are of non-zero digits are notnot significant significant

Act as placeholdersAct as placeholdersCan’t be dropped, show magnitudeCan’t be dropped, show magnitude

Examples:Examples:0.00024 = 20.00024 = 20.453 = 0.453 = 3

Rule #4 – Trailing 0’s with DPRule #4 – Trailing 0’s with DP

Zeros at the Zeros at the endend of a number of a number andand to the to the rightright of of a decimal point are significant.a decimal point are significant.

Examples:Examples:

43.00 = 443.00 = 4

1.010 = 41.010 = 4

1.50 =1.50 =3

Rule #5 – Trailing 0’s without DPRule #5 – Trailing 0’s without DP

Zeros at the Zeros at the endend of a number of a number andand to the to the leftleft of a of a decimal point decimal point aren’taren’t significant significant

Examples:Examples:

300 = 1300 = 1

27,300 = 27,300 = 3

Interesting Answers to Catholic Elementary School Bible Questions:

Many religions teach that you are to have only one wife

This is called Monotony.

Easier Way to do Sig Figs!!Easier Way to do Sig Figs!!

Pacific/AtlanticPacific/Atlantic

P A

If a decimal point is present, start on the Pacific (P) side and draw an arrow through the number until you hit a non-zero digit. Count all

numbers without an arrow through them.

If a decimal is absent, start on the Atlantic (A) side and draw an arrow through the number until you hit a non-zero digit.

Examples:Examples:

123.003 grams123.003 grams

decimal present, start on “P” side, draw arrow, decimal present, start on “P” side, draw arrow, count digits without an arrow through it.count digits without an arrow through it.

Answer = 6Answer = 6

10,100 centimeters10,100 centimeters

Decimal absent, start on “A” side, draw an arrow, Decimal absent, start on “A” side, draw an arrow, count digits without an arrow through it.count digits without an arrow through it.

Answer = 3Answer = 3


A. Which answer(s) contain 3 significant figures?A. Which answer(s) contain 3 significant figures?

1) 0.47601) 0.4760 2) 0.00476 2) 0.00476 3) 4760 3) 4760

B. All the zeros are significant inB. All the zeros are significant in

1) 0.00307 1) 0.00307 2) 25.300 2) 25.300 3) 2.050 x 10 3) 2.050 x 1033

C. 534,675 rounded to 3 significant figures isC. 534,675 rounded to 3 significant figures is

1) 535 1) 535 2) 535,000 2) 535,000 3) 5.35 x 10 3) 5.35 x 1055


In which set(s) do both numbers contain the In which set(s) do both numbers contain the samesame number of significant figures?number of significant figures?

1) 22.0 and 22.00 1) 22.0 and 22.00

2) 400.0 and 40 2) 400.0 and 40

3) 0.000015 and 150,0003) 0.000015 and 150,000

Significant Figures and Numbers

Some numbers are exact: There are60 seconds in 1

minute

25 cents in 1 quarter

12 eggs in one dozen

There is no uncertainty in any of these numbers.

In other words there are

12.0000000000000000000000000000000000 eggs in 1 dozen

(add as many zeros as you like)

Counting NumbersCounting Numbers

Counting numbers have infinite sig figs.Counting numbers have infinite sig figs.

Ex: 3 apples Ex: 3 apples

= 3.000 000 000 000 … apples= 3.000 000 000 000 … apples

Significant FiguresSignificant FiguresLets’ see if you can:Lets’ see if you can:

1. Explain what significant figures are in a measurement

2. Determine the number of significant figures in any measurement

State the number of significant figures in each State the number of significant figures in each of the following:of the following:

A. 0.030 mA. 0.030 m 1 1 2 2 3 3

B. 4.050 LB. 4.050 L 2 2 3 3 4 4

C. 0.0008 gC. 0.0008 g 1 1 2 2 4 4

D. 3.00 mD. 3.00 m 1 1 2 2 3 3

E. 2,080,000 beesE. 2,080,000 bees 3 3 5 5 7 7



And Mathematical CalculationsAnd Mathematical Calculations

Significant FiguresSignificant FiguresAt the conclusion of our time At the conclusion of our time

together, you should be able to:together, you should be able to:

1. Determine the number of significant figures needed for an answer involving calculations.

2. Round math problems properly

13

Rules for Rounding OffRules for Rounding Off

If the digit to be removedIf the digit to be removed• is less than 5, the preceding digit stays is less than 5, the preceding digit stays

the samethe same• is equal to or greater than 5, the is equal to or greater than 5, the

preceding digit is increased by 1preceding digit is increased by 1 In a series of calculations, carry the extra In a series of calculations, carry the extra

digits to the final result and digits to the final result and thenthen round off round off Don’t forget to add place-holding zeros if Don’t forget to add place-holding zeros if

necessary to keep value the same!!necessary to keep value the same!!

Significant Figure Math RulesSignificant Figure Math Rules

Addition / Subtraction Problem:Addition / Subtraction Problem:Penny Example = Penny Example = 0.010.0199 m using meter stick m using meter stick

0.0190.01922 m using ruler m using ruler0.0191 m using calipers0.0191 m using calipers0.019040.0190466 m using m using

micrometermicrometerTo find the total = 0.076346 mTo find the total = 0.076346 m

But most of my measurements have fewer But most of my measurements have fewer decimal places than my best tool!!!decimal places than my best tool!!!


Addition / Subtraction:Addition / Subtraction:Answers can’t have more numbers Answers can’t have more numbers

to the right of the decimal point than the to the right of the decimal point than the number in the problem with the number in the problem with the leastleast amount of numbers to the right of the amount of numbers to the right of the decimal point.decimal point.

Example = Example = 24.24.11 m + 3.35 m + 2.23 m + 3.35 m + 2.23 mm

Calculator says:Calculator says: 29.68 m (wrong)29.68 m (wrong)Answer:Answer: 29.29.77 m m

Another Example of Adding and Another Example of Adding and SubtractingSubtracting

The answer has the same number of decimal The answer has the same number of decimal places as the measurement with the fewest places as the measurement with the fewest decimal places.decimal places.

25.25.22 m m one decimal placeone decimal place

+ 1.+ 1.3434 m m two decimal placestwo decimal places

26.54 m26.54 m

answer 26.5 answer 26.5 m m one decimal placeone decimal place


Let’s Try a Multiplication / Division Let’s Try a Multiplication / Division Problem: Find the volume?Problem: Find the volume?

0.040.0411m highm high

0.090.0911 m wide m wide

0.030.0344 m deep m deep

0.0000.00012691269 m m33

What should my answer be??What should my answer be??


Multiplication / Division Problem:Multiplication / Division Problem:

14.14.11 cm cm

3.3.33 cm cm

4.234.23 cm cm22

42.42.33 cm cm22

446.536.53 cm cm22



Multiplication / Division:Multiplication / Division:

Your answer can’t have more sig figs than the Your answer can’t have more sig figs than the number in the problem with the number in the problem with the leastleast amount amount of sig figsof sig figs

ExampleExample = = 60.56227892 cm x 60.56227892 cm x 35.2535.25 cm cm

Calculator says: Calculator says: 2134.890832 cm2134.890832 cm22 (wrong) (wrong)

Answer: Answer: 21352135 cm cm22

Chemical Compound QuizChemical Compound Quiz

Which is denser: ice or water?

Water

Why??

Ice Expands 1/7th!!!

Significant FiguresSignificant FiguresLets’ see if you can:Lets’ see if you can:

1. Determine the number of significant figures needed for an answer involving calculations.

2. Round math problems properly


Remember this Problem:Remember this Problem:Penny Example = Penny Example = 0.010.0199 m using meter stick m using meter stick

0.0190.01922 m using ruler m using ruler0.0191 m using calipers0.0191 m using calipers0.019040.0190466 m using m using

micrometermicrometerTo find the total = 0.076346 mTo find the total = 0.076346 m

0.076 m


Remember This One:Remember This One:

14.14.11 cm cm

3.3.33 cm cm

4.234.23 cm cm22

42.42.33 cm cm22

446.536.53 cm cm22


47 cm47 cm22


How About This One: How About This One: Find the volume? Find the volume?

0.040.0411m highm high

0.090.0911 m wide m wide

0.030.0344 m deep m deep

0.0000.00012691269 m m33


0.00013 m0.00013 m33


1. 2.19 m X 4.2 m = 1. 2.19 m X 4.2 m = A) 9 mA) 9 m22 B) 9.2 m B) 9.2 m22 C) 9.198 m C) 9.198 m22

2. 4.311 m ÷ 0.07 m = 2. 4.311 m ÷ 0.07 m = A)A) 61.5861.58 B) 62 B) 62 C) 60 C) 60

3. 3. 2.54 m X 0.0028 m2.54 m X 0.0028 m = =

0.0105 m X 0.060 m0.0105 m X 0.060 m

A) 11.3 B) 11 A) 11.3 B) 11 C) 10 C) 10


In each calculation, round the answer to the In each calculation, round the answer to the correct number of significant figures.correct number of significant figures.

1. 235.05 m + 19.6 m + 2.1 m = 1. 235.05 m + 19.6 m + 2.1 m =

A) 256.75 mA) 256.75 m B) 256.8 mB) 256.8 m C) 257 m C) 257 m

2. 58.925 m - 18.2 m2. 58.925 m - 18.2 m ==

A) 40.725 mA) 40.725 m B) 40.73 m C) 40.7 mB) 40.73 m C) 40.7 m

Euphemisms in ScienceEuphemisms in Science We all know that some politicians and government We all know that some politicians and government

spokesmen use certain euphemistic phrases to give an spokesmen use certain euphemistic phrases to give an aura of respectability to descriptions of events or actions aura of respectability to descriptions of events or actions

which would be offensive when expressed in plain English. which would be offensive when expressed in plain English. The following is a list of Euphemisms in Science and their The following is a list of Euphemisms in Science and their

translations into plain English.translations into plain English.

“Handled with extreme care during the experiments...”

Not dropped on the floor!

significant figures

Documents

number of significant

cmsecond digit

additional estimated

given measurement

nonzero digit

cmthird digit

total number of digits

insignificant digits